Georgia Institute of Technology College of Sciences

Random effects models are commonly used to measure genetic variance-covariance matrices of quantitative phenotypic traits. The population eigenvalues of these matrices describe the evolutionary response to selection. However, they may be difficult to estimate from limited samples when the number of traits is large. In this talk, I will present several results describing the eigenvalues of classical MANOVA estimators of these matrices, including dispersion of the bulk eigenvalue distribution, bias and aliasing of large "spike" eigenvalues, and distributional limits of eigenvalues at the spectral edges. I will then discuss a new procedure that uses these results to obtain better estimates of the large population eigenvalues when there are many traits, and a Tracy-Widom test for detecting true principal components in these models. The theoretical results extend proof techniques in random matrix theory and free probability, which I will also briefly describe.This is joint work with Iain Johnstone, Yi Sun, Mark Blows, and Emma Hine.